Question

设F'(x)=f(x) d/dx∫(下限a上限b)f(x+y)dy =d(F(b+x)-F(a+x))/dx 怎么来的

Answer 1

令u = x + y、du = dy
∫(a→b) f(x + y) dy
当y = a、u = x + a
当y = b、u = x + b
变为∫(x + a→x + b) f(u) du
所以d/dx ∫(a→b) f(x + y) dy
= d/dx ∫(x + a→x + b) f(u) du
= d(x + b)/dx * f(x + b) - d(x + a)/dx * f(x + a)、公式d/dx ∫(0→x) f(t) dt = f(x)
= dF(x + b)/dx - dF(x + a)/dx、链式法则d/dx F(g(x)) = dF(g(x))/d[g(x)] * d[g(x)]/dx = dF(u)/du * du/dx
= d[F(x + b) - F(x + a)]/dx、这里F(g(x)) = F(x + b) = F(u)、u = g(x) = x + b